Patch bays come with two rows of jacks. Most often the top row will be outputs, and the bottom row is inputs. A normalled connection will send the signal from the top row to the bottom row without anything being plugged in. With a normalled connection, if you plug in a patch cable, the normalled connection is broken. With a half normalled patch bay, the connection is only broken when you plug in a connection to the bottom row (the input).
The idea is that you can have the units you use most connected without having to use any patch cables. For instance, you might keep the input to your reverb unit "normalled" to the first aux send on your console, and the output "normalled" to a pair of channels. This saves a lot of patch cables, which is very important because (1) patch cables are relatively expensive, and (2) it can be very difficult to figure out what is plugged in where, and where to plug it, when there are so many cables in the way.
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